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Models for Circular–Linear and Circular–Circular Data Constructed from Circular Distributions Based on Nonnegative Trigonometric Sums

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Johnson and Wehrly (1978, Journal of the American Statistical Association73, 602–606) and Wehrly and Johnson (1980, Biometrika67, 255–256) show one way to construct the joint distribution of a circular and a linear random variable, or the joint distribution of a pair of circular random variables from their marginal distributions and the density of a circular random variable, which in this article is referred to as joining circular density. To construct flexible models, it is necessary that the joining circular density be able to present multimodality and/or skewness in order to model different dependence patterns. Fernández-Durán (2004, Biometrics60, 499–503) constructed circular distributions based on nonnegative trigonometric sums that can present multimodality and/or skewness. Furthermore, they can be conveniently used as a model for circular–linear or circular–circular joint distributions. In the current work, joint distributions for circular–linear and circular–circular data constructed from circular distributions based on nonnegative trigonometric sums are presented and applied to two data sets, one for circular–linear data related to the air pollution patterns in Mexico City and the other for circular–circular data related to the pair of dihedral angles between consecutive amino acids in a protein.

Keywords: Circular distributions; Circular–circular data; Circular–linear data; Trigonometric sums

Document Type: Research Article


Affiliations: Department of Statistics, Instituto Tecnológico Autónomo de México, Río Hondo No. 1, Col. Tizapán San Angel, C.P. 01000, México D.F., México, Email:

Publication date: June 1, 2007

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